Final answer:
To show a function is a linear transformation, ensure it preserves both vector addition and scalar multiplication. Represent linear equations graphically as straight lines described by y = b + mx to visualize the transformation.
Step-by-step explanation:
To show that a function is a linear transformation, two main properties must hold. First, the function must preserve vector addition, meaning that the image of the sum of two vectors should be the sum of the images of those vectors. Second, the function must preserve scalar multiplication, meaning that the image of a scalar multiple of a vector should be the same scalar multiple of the image of that vector.
These two properties can be formally expressed as:
- If f is a linear transformation, then f(u + v) = f(u) + f(v) for all vectors u and v in the domain.
- If f is a linear transformation, then f(cu) = cf(u) for all vectors u in the domain and all scalars c.
Graphically, expressing equations graphically, a linear equation can be represented in the form y = b + mx, which describes a straight line on a graph. The slope (m) determines the steepness of the line, and the y-intercept (b) specifies the point at which the line crosses the y-axis. Expressing relationships as linear equations is fundamental when studying linear transformations as it helps in understanding how changes in input (x) affect outputs (y).